Rationality of the moduli spaces of plane curves of sufficiently large degree

Böhning, Christian; Bothmer, Hans-Christian Graf von

doi:10.1007/s00222-009-0214-6

Cited by 3 publications

(4 citation statements)

References 14 publications

(21 reference statements)

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“…Proof. Follows from Theorem 2.16, Theorem 2.21 and results on rationality of plane curves in [BG10] and [BGK09] .…”

Section: Tautological Family On Zariski Open Sets Of Coarse Moduli An...mentioning

confidence: 85%

Tautological families of cyclic covers of projective spaces

Kundu¹,

Mukherjee²,

Raychaudhury³

2021

Preprint

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In this article, we study the existence of tautological families on a Zariski open set of the coarse moduli space parametrizing certain Galois covers over projective spaces. More specifically, let (1) H n.r.d (resp. M n,r,d ) be the stack (resp. coarse moduli) parametrizing smooth simple cyclic covers of degree r over the projective space P n branched along a divisor of degree r d ≥ 4, and (2) d 2 ) be the stack (resp. coarse moduli) of smooth cyclic triple covers over P 1 with 2d 1 − d 2 ≥ 4 and 2d 2 − d 1 ≥ 4. In the former case, we show that such a family exists if and only if gcd(r d,n +1) | d while in the latter case, we show that it always exists. We further show that even when such a family exists, often it cannot be extended to the open locus of objects without extra automorphisms. The existence of tautological families on a Zariski open set of its coarse moduli can be interpreted in terms of rationality of the stack if the coarse moduli space is rational. Combining our results with known results on the rationality of the coarse moduli of points on P 1 and the coarse moduli of plane curves (see [BG10], [BGK09]), we determine the rationality of H 1,r,d (resp. H 2,r,d ) for r d ≥ 4 (resp. r d ≥ 49). On the other hand H 1,3,d 1 ,d 2 is unirational, and we show that its coarse moduli M 1,3,d 1 ,d 2 is unirational and fibred over a rational base by homogeneous varieties which are rational if char(k) = 0. Our study is motivated by [GV09] and [GV08].

show abstract

“…Proof. Follows from Theorem 2.16, Theorem 2.21 and results on rationality of plane curves in [BG10] and [BGK09] .…”

Section: Tautological Family On Zariski Open Sets Of Coarse Moduli An...mentioning

confidence: 85%

Tautological families of cyclic covers of projective spaces

Kundu¹,

Mukherjee²,

Raychaudhury³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Acting by SL 3 (C) we may suppose that these points are the four vertices of a frame of reference in P 2 and so SL 3 (C) acts with a dense orbit in Grass(2, Sym 2 (V )). The stabilizer of a generic point in Grass(2, Sym 2 (V )) inside PSL 3 4 , where F is the generic fibre of ϕ, is birational to M B by Corollary 2.7 and the preceding discussion. We will determine F as S 4 -representation.…”

mentioning

confidence: 85%

“…. , l 5 ∈ (C 3 ) ∨ such that C = {l 4 1 + · · · + l 4 5 = 0}. As in the previous item we denote by P C the subset of Clebsch quartics inside the space P 14 of all plane quartics.…”

Section: Lüroth Quartics and Bateman Pointsmentioning

confidence: 99%